(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 206499, 4218] NotebookOptionsPosition[ 203680, 4128] NotebookOutlinePosition[ 204038, 4144] CellTagsIndexPosition[ 203995, 4141] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["MAT 331:", FontSize->24], " ", StyleBox["Project 1", FontSize->24] }], "Section", CellChangeTimes->{{3.618884780257655*^9, 3.618884795191327*^9}, { 3.619013741815179*^9, 3.619013742915434*^9}, {3.6190639773624067`*^9, 3.6190639931423893`*^9}, 3.6190640316767807`*^9, {3.620115017220894*^9, 3.620115018260913*^9}, {3.620271183359037*^9, 3.620271185480665*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Project description:", "Subsection"]], "Section", CellChangeTimes->{{3.61895751446417*^9, 3.61895752344552*^9}, { 3.620185321017673*^9, 3.620185321160325*^9}, {3.620275496470414*^9, 3.620275501630445*^9}, {3.620279164310918*^9, 3.620279164805422*^9}}], Cell["\<\ The PageRank algorithm provides the basis for Google\[CloseCurlyQuote]s web \ search tools. As we described in the lecture, PageRank is a recursive \ algorithm, and it works by counting the number and quality of links to a page \ to determine a rough estimate of how important the website is. The underlying \ assumption is that more important websites are likely to receive more links \ from other websites and therefore, the importance of any website can be \ judged by looking at the number and quality of the pages that link to it. \ Components of the PageRank vector serve as \[OpenCurlyDoubleQuote]importance\ \[CloseCurlyDoubleQuote] weights for web pages, independent of their textual \ content, solely based on the hyperlink structure of the web. In this project we are interested in the theoretical foundations of the \ PageRank algorithm, in computing the Page Rank of small-scale graphs and in \ the effects of particular aspects of web graph structure on the Page Rank \ vector. \ \>", "Text", CellChangeTimes->{{3.618958035675892*^9, 3.61895825489732*^9}, { 3.61895828758503*^9, 3.618958309439242*^9}, {3.6190090575312862`*^9, 3.619009060737265*^9}, {3.619009133231318*^9, 3.6190091577092857`*^9}, { 3.619009297082733*^9, 3.619009305010374*^9}, {3.6190531597810783`*^9, 3.619053163786626*^9}, {3.619057974295499*^9, 3.619058008308049*^9}, { 3.619063099075738*^9, 3.619063221510716*^9}, {3.620104581154149*^9, 3.620104624460071*^9}, 3.620104952514773*^9, {3.620105306189464*^9, 3.62010549225408*^9}, {3.620106757254405*^9, 3.620106783973249*^9}, { 3.62010690214373*^9, 3.6201069404389887`*^9}, {3.620107003675831*^9, 3.620107046135168*^9}, {3.6201072140945473`*^9, 3.6201072521827602`*^9}, { 3.620107296382296*^9, 3.620107686022007*^9}, {3.6201087715394707`*^9, 3.620108772193202*^9}, {3.620108906139463*^9, 3.620108911090001*^9}, { 3.620108943078869*^9, 3.620108945423254*^9}, 3.620113287306345*^9, { 3.620113350538128*^9, 3.620113413916574*^9}, {3.620113448356659*^9, 3.620113599560532*^9}, {3.620114665213971*^9, 3.620114736951397*^9}, { 3.6201148474850683`*^9, 3.6201148752584467`*^9}, {3.620114909764741*^9, 3.620114911396558*^9}, {3.6201149523365803`*^9, 3.62011500488629*^9}, { 3.620189964186058*^9, 3.620189986023975*^9}, {3.620190318682701*^9, 3.6201903257366133`*^9}, {3.620190423660921*^9, 3.620190427915267*^9}, { 3.620197347691531*^9, 3.620197365990987*^9}, {3.6201974043855877`*^9, 3.620197406994176*^9}, {3.6202719954290123`*^9, 3.620271996680101*^9}, { 3.620272255839624*^9, 3.620272304959227*^9}, {3.620272367183546*^9, 3.620272377230268*^9}, {3.620272431118443*^9, 3.620272434436957*^9}, { 3.620272515590114*^9, 3.6202725196350317`*^9}, {3.620272624593026*^9, 3.620272640079751*^9}, {3.620272766275238*^9, 3.620272767296754*^9}, { 3.620272813281234*^9, 3.620272834730152*^9}, {3.620272921996311*^9, 3.6202729625233593`*^9}, {3.620273088967372*^9, 3.6202731231558027`*^9}, { 3.620273153176917*^9, 3.620273157862782*^9}, {3.620273193719795*^9, 3.6202732512557907`*^9}, {3.620273416186001*^9, 3.620273418256668*^9}, { 3.620274180406227*^9, 3.620274227030546*^9}, {3.62027443218283*^9, 3.620274467439011*^9}, {3.620275453349742*^9, 3.620275483006337*^9}, { 3.6209528460864153`*^9, 3.620952865759512*^9}}, TextJustification->1.] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Tasks:", "Subsection"]], "Section", CellChangeTimes->{{3.6188834470493517`*^9, 3.618883458254191*^9}, { 3.6188841737965927`*^9, 3.618884177712782*^9}, {3.6188846295181627`*^9, 3.6188846297983932`*^9}, {3.619056965231044*^9, 3.619056965742797*^9}, { 3.620185345872222*^9, 3.620185347792323*^9}, {3.620275634697048*^9, 3.620275638063816*^9}}], Cell[CellGroupData[{ Cell["\<\ Browse the web and read about the importance of the PageRank algorithm for \ web searches. Ranking algorithms such as PageRank have also found \ applications in other areas (population models, biology, article citations, \ hotel ranking systems, etc.). Give concrete examples of the applications of \ Page Rank to search engines and to at least three other real life problems. \ In each case, briefly describe how the ranking algorithm is applied. Please \ include all relevant references.\ \>", "ItemNumbered", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274002206958*^9}, {3.620274040192863*^9, 3.620274040195754*^9}, {3.6202753291045303`*^9, 3.620275367535472*^9}, { 3.620275884759132*^9, 3.620275989623373*^9}, {3.620276021928265*^9, 3.6202760459204493`*^9}, {3.620277460106645*^9, 3.620277482904023*^9}, { 3.6202785983826103`*^9, 3.620278603068809*^9}, {3.620289020739518*^9, 3.620289027106184*^9}, {3.6526642807522907`*^9, 3.652664322513302*^9}, { 3.652664352963787*^9, 3.652664353933444*^9}, {3.6970445576816397`*^9, 3.6970446756002493`*^9}}, TextJustification->1., FontSize->14], Cell["\<\ Describe how the Page Rank algorithm works. Investigate possible ways to \ compute the Page Rank vector of a graph (that is, computing eigenvalues, or \ rather applying the Power Method) and describe their advantages and drawbacks \ as the size of the graph gets bigger and bigger. \ \>", "ItemNumbered", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274002206958*^9}, {3.620274040192863*^9, 3.620274053329183*^9}, {3.620274860937676*^9, 3.620274926637228*^9}, { 3.6202749851904182`*^9, 3.6202749878384113`*^9}, {3.620275794800355*^9, 3.620275862654866*^9}, {3.6202760659606743`*^9, 3.620276067528509*^9}, { 3.65266440629994*^9, 3.652664410427711*^9}, {3.652666227569915*^9, 3.652666244060054*^9}, 3.6526663139358892`*^9}, TextJustification->1., FontSize->14], Cell["\<\ Explain the mathematics behind PageRank. The algorithm can be explained \ using ideas from several fields of Mathematics, like probability, linear \ algebra, dynamical systems or analysis. You can choose the mathematical \ language that you feel more comfortable with. Feel free to use external \ sources and articles and include references to any relevant theorems and \ proofs that you may decide to use. If you decide to follow the steps \ explained in the lecture notes, then take a look at the lemmas listed at the \ end and fill in the proofs.\ \>", "ItemNumbered", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274024316057*^9}, {3.620274061800963*^9, 3.620274156089809*^9}, {3.620274569781389*^9, 3.6202746696482286`*^9}, { 3.6202747043842173`*^9, 3.620274710016069*^9}, {3.620275168089028*^9, 3.6202752488831577`*^9}, {3.6202760722488947`*^9, 3.6202760762404222`*^9}, {3.620276110310635*^9, 3.620276126352111*^9}, 3.6202776894069443`*^9, {3.620873826232725*^9, 3.620873859766376*^9}, { 3.6208787424066353`*^9, 3.6208787773443213`*^9}, {3.620880588479624*^9, 3.6208805923983803`*^9}, {3.620882106832362*^9, 3.62088212246988*^9}, { 3.652664434182571*^9, 3.6526644407123823`*^9}, {3.697044710520392*^9, 3.6970447344075203`*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "You will next build an interactive model in ", StyleBox["Mathematica", FontSlant->"Italic"], " for calculating the Page Rank vector of small scale web graphs, with nodes \ representing websites and edges representing hyperlinks. The number of nodes \ is assumed reasonably small, so that your algorithm works relatively fast. \ Your model should work with ", StyleBox["random graphs", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["graphs with dangling nodes", FontColor->RGBColor[0.5, 0, 0.5]], ", as well as some fixed ", StyleBox["test graphs", FontColor->RGBColor[1, 0.5, 0]], ", given below. The basic model should display the graph with the first five \ most important nodes highlighted in different colors and with increased \ sizes, and the (modified?) transition matrix A of the graph. The extended \ model should include a step-by step computation of the page-rank vector (at \ each step, one computes ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["A", "k"], "v"}], TraditionalForm]]], " and highlights the most important 5 vertices according to the page-rank \ approximation ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["A", "k"], "v"}], TraditionalForm]]], "). There are several aspects that your interactive model should take into \ account:" }], "ItemNumbered", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274024316057*^9}, {3.620274061800963*^9, 3.620274082225349*^9}, {3.620274339132412*^9, 3.620274379121662*^9}, { 3.620274514088357*^9, 3.620274534270507*^9}, {3.620276170532715*^9, 3.620276273792305*^9}, {3.620276304110758*^9, 3.620276319115917*^9}, { 3.620276371807721*^9, 3.6202765942195473`*^9}, {3.6202768687132387`*^9, 3.620276868716264*^9}, {3.620277227143683*^9, 3.62027724734381*^9}, { 3.620277282907153*^9, 3.6202772929955893`*^9}, {3.620277329937893*^9, 3.620277369415407*^9}, {3.620277567016253*^9, 3.620277569264638*^9}, { 3.620278229020762*^9, 3.620278229023822*^9}, {3.6202786899944963`*^9, 3.6202786964835463`*^9}, {3.620278730930043*^9, 3.620278734889879*^9}, { 3.620278880923047*^9, 3.620279061859725*^9}, {3.620279102809157*^9, 3.6202791072874928`*^9}, {3.6209498315215893`*^9, 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3.620954117687533*^9}, TextJustification->1., FontSize->14], Cell["\<\ The damping factor p should be given as a parameter, with values between 0.01 \ and 0.5. \ \>", "Subitem", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274024316057*^9}, {3.620274061800963*^9, 3.620274082225349*^9}, {3.620274339132412*^9, 3.620274379121662*^9}, { 3.620274514088357*^9, 3.620274534270507*^9}, {3.620276170532715*^9, 3.620276273792305*^9}, {3.620276304110758*^9, 3.620276319115917*^9}, { 3.620276371807721*^9, 3.6202765942195473`*^9}, {3.6202768687132387`*^9, 3.620276868716264*^9}, {3.620277227143683*^9, 3.62027724734381*^9}, { 3.620277282907153*^9, 3.6202772929955893`*^9}, {3.620277329937893*^9, 3.620277369415407*^9}, {3.620277567016253*^9, 3.620277569264638*^9}, { 3.620278229020762*^9, 3.620278229023822*^9}, {3.6202786899944963`*^9, 3.6202786964835463`*^9}, {3.620278730930043*^9, 3.620278734889879*^9}, { 3.620278880923047*^9, 3.620279061859725*^9}, {3.620279102809157*^9, 3.6202791072874928`*^9}, {3.6209498315215893`*^9, 3.620949954060083*^9}, { 3.620950106261961*^9, 3.620950168436288*^9}, {3.6209529256263847`*^9, 3.620952927808035*^9}, {3.62095297933768*^9, 3.620953156044487*^9}, { 3.6209533217751102`*^9, 3.6209533399278927`*^9}, {3.6209533732512283`*^9, 3.620953416155357*^9}, {3.620953484419508*^9, 3.620953607846266*^9}, { 3.620953718216112*^9, 3.6209537462861443`*^9}, {3.6209616449529*^9, 3.6209616455195427`*^9}, {3.620961684902123*^9, 3.6209616867338123`*^9}, { 3.652664533194334*^9, 3.652664533860134*^9}, 3.65266457926882*^9}, TextJustification->1., FontSize->14], Cell["\<\ The power k should also be given as a parameter, with values between 1 and \ 50, and with step size 1.\ \>", "Subitem", CellChangeTimes->CompressedData[" 1:eJwlx08oQwEAx/GXrak3mmYOyr9pJOGykZXoOZiyA3KQ2sHMn/0RraSVEW35 lzhIyJp/tY0DqylykT/LxdTjgK0lduCNopa2kbzfc/j26SvVDrR2pxAEkcMG ZSOi7fwEQ6nzhjhtHdfnUJZLvkPrxtEnFFWpYvCwRicpYHVHurIhfVNfCn9d 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3.6202772929955893`*^9}, {3.620277329937893*^9, 3.620277369415407*^9}, {3.620277567016253*^9, 3.620277569264638*^9}, { 3.620278229020762*^9, 3.620278302791202*^9}, {3.620278641996458*^9, 3.6202786435161858`*^9}}, TextJustification->1., FontSize->14], Cell["\<\ Conclusions. Formulate your findings after testing your interactive model on \ a couple of examples. Is the Page Rank vector in agreement with your natural \ understanding of the relative importance of nodes? What values of the damping \ factor seem to give more accurate rankings? You can also copy paste in here \ any interesting examples that you may have discovered when testing your model.\ \>", "ItemNumbered", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274024316057*^9}, {3.620274061800963*^9, 3.620274082225349*^9}, {3.620274339132412*^9, 3.620274379121662*^9}, { 3.620274514088357*^9, 3.620274534270507*^9}, {3.620276170532715*^9, 3.620276273792305*^9}, {3.620276304110758*^9, 3.620276319115917*^9}, { 3.620276371807721*^9, 3.6202765942195473`*^9}, {3.6202768687132387`*^9, 3.620276888906393*^9}, {3.620276922543928*^9, 3.6202771665980263`*^9}, { 3.620278162943123*^9, 3.620278186045005*^9}, {3.620880549284628*^9, 3.6208805513561707`*^9}, 3.697045056278913*^9}, TextJustification->1., FontSize->14], Cell["References. Include all relevant references here.", "ItemNumbered", CellChangeTimes->{{3.620273440483904*^9, 3.6202734436599483`*^9}, { 3.6202737024797993`*^9, 3.620274024316057*^9}, {3.620274061800963*^9, 3.620274082225349*^9}, {3.620274339132412*^9, 3.620274379121662*^9}, { 3.620274514088357*^9, 3.620274534270507*^9}, {3.620276170532715*^9, 3.620276273792305*^9}, {3.620276304110758*^9, 3.620276319115917*^9}, { 3.620276371807721*^9, 3.6202765942195473`*^9}, {3.6202768687132387`*^9, 3.620276917840207*^9}, {3.620277175221816*^9, 3.620277180173583*^9}, { 3.620277533241397*^9, 3.620277534112125*^9}, {3.620880534708021*^9, 3.620880537012148*^9}}, TextJustification->1., FontSize->14] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["General considerations:", "Subsection"]], "Section", CellChangeTimes->{{3.6188834470493517`*^9, 3.618883458254191*^9}, { 3.6188841737965927`*^9, 3.618884177712782*^9}, {3.6188846295181627`*^9, 3.6188846297983932`*^9}, {3.619056965231044*^9, 3.619056965742797*^9}, { 3.620184812387269*^9, 3.620184812898695*^9}, {3.620185327776074*^9, 3.620185327888328*^9}, {3.6202755381324787`*^9, 3.620275543299217*^9}, 3.6202756456554832`*^9}], Cell[TextData[{ "You will be graded on the quality and maturity of your mathematical \ explanations, as well as on the efficiency of the ", StyleBox["Mathematica", FontSlant->"Italic"], " code that you develop. Your interactive model should reevaluate itself \ only when the controls are changed (i.e. we change the position of a slider, \ etc.). When possible, try to avoid performing unnecessary computations (for \ example, do not use Eigensystem[A] if you are only interested in the \ eigenspace corresponding to the eigenvalue 1; you may choose not to use \ MatrixPower[A,k] if you have to compute ", Cell[BoxData[ FormBox[ SuperscriptBox["A", "k"], TraditionalForm]]], " for many successive values of k; etc.) Try to have a nice coding style and \ use functions when appropriate. Use different names for your variables! Do \ not include all intermediate output that you may have obtained by testing \ small parts of your code. All your code, including functions/variable \ definitions and Manipulate[..] should be contained in one single ", StyleBox["Mathematica", FontSlant->"Italic"], " cell bracket. Before you start working on your project, look also through \ the ", ButtonBox["Wolfram Demonstration Projects,", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/"], None}, ButtonNote->"http://demonstrations.wolfram.com/"], " to get an idea of how a project should look like." }], "Text", CellChangeTimes->{{3.6190083437795057`*^9, 3.619008619778887*^9}, { 3.6190086997717037`*^9, 3.619008742318366*^9}, {3.61900877337542*^9, 3.619008846690468*^9}, {3.619063502179244*^9, 3.619063504155314*^9}, { 3.619064140835752*^9, 3.619064155080373*^9}, {3.6194770073674107`*^9, 3.6194770491064157`*^9}, {3.620075409161674*^9, 3.620075433432282*^9}, { 3.620075524069899*^9, 3.620075731127187*^9}, {3.620075775367186*^9, 3.6200758388256407`*^9}, {3.620075869795953*^9, 3.620076143652358*^9}, { 3.620076194766493*^9, 3.620076199750688*^9}, {3.620184195501205*^9, 3.6201842312101793`*^9}, {3.620184266043998*^9, 3.620184324595572*^9}, { 3.620184405840395*^9, 3.620184408351913*^9}, 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Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematics:", "Subsection"]], "Section", CellChangeTimes->{{3.6188834470493517`*^9, 3.618883458254191*^9}, { 3.6188841737965927`*^9, 3.618884177712782*^9}, {3.6188846295181627`*^9, 3.6188846297983932`*^9}, {3.619056965231044*^9, 3.619056965742797*^9}, { 3.620184812387269*^9, 3.620184812898695*^9}, {3.620185327776074*^9, 3.620185327888328*^9}, {3.6202755381324787`*^9, 3.620275543299217*^9}, 3.6202756456554832`*^9, {3.62087415940935*^9, 3.620874181191745*^9}}], Cell["\<\ When explaining the Mathematics behind the Page Rank Algorithm, if you decide \ to follow the plan outlined in the lectures, don\[CloseCurlyQuote]t forget to \ give justifications for the following statements:\ \>", "Text", CellChangeTimes->{{3.620874193191876*^9, 3.620874222623838*^9}, { 3.6208787052185183`*^9, 3.620878711673015*^9}, {3.620880397993547*^9, 3.620880437328911*^9}, {3.620881797230989*^9, 3.62088184246094*^9}}], Cell[TextData[{ "If A is a column stochastic nxn matrix and v\[Element]", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], " is a probabilistic vector, then Av is a probabilistic vector." }], "SubitemNumbered", CellChangeTimes->{{3.62087425935975*^9, 3.620874292265623*^9}, { 3.620875214531196*^9, 3.6208752231264563`*^9}, {3.620875259504925*^9, 3.6208752896320467`*^9}, {3.6526648264967117`*^9, 3.652664828759376*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Hint: By the definition of matrix multiplication, the i-th entry \ of the vector Av is ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{"(", "Av", ")"}], "i"], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox[" = ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ StyleBox[ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "1"}], "n"], FontSize->18], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "ij"], SubscriptBox["v", "j"]}], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox[".", FontSlant->"Italic", FontColor->GrayLevel[0.5]] }], "Item", CellChangeTimes->{{3.620952522116192*^9, 3.620952732401578*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "If A is a positive nxn matrix and v\[Element]", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], " a probabilistic vector with non-negative entries, then Av is a vector with \ all entries positive." }], "SubitemNumbered", CellChangeTimes->{{3.62087425935975*^9, 3.62087432819341*^9}, { 3.620875231619055*^9, 3.620875233026581*^9}, {3.620875275925272*^9, 3.620875282014454*^9}, {3.620876538578767*^9, 3.620876538580887*^9}, { 3.654028114015225*^9, 3.654028117175579*^9}}], Cell[TextData[{ "If A is a column stochastic nxn matrix, B is the nxn matrix with all \ entries equal to 1 and p\[Element](0,1], then the matrix (1-p)A + p", Cell[BoxData[ FormBox[ FractionBox["1", "n"], TraditionalForm]], FontSize->16], "B is positive and column stochastic. The number p is called the damping \ factor." }], "SubitemNumbered", CellChangeTimes->{{3.62087425935975*^9, 3.62087432819341*^9}, { 3.620875231619055*^9, 3.620875233026581*^9}, {3.620875275925272*^9, 3.620875282014454*^9}, {3.620876538578767*^9, 3.620876808507765*^9}}], Cell[TextData[{ "Let u and v be two probabilistic vectors in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], ". Define the distance between u and v to be ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}]}], "=", FractionBox["1", "2"], " "}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "n"], FontSize->18], RowBox[{"|", RowBox[{ SubscriptBox["u", "i"], "-"}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]]], "|. " }], "SubitemNumbered", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208748009783773`*^9}, {3.620875123410638*^9, 3.62087512396216*^9}, 3.620875204216419*^9, {3.620875302089512*^9, 3.620875315896203*^9}, { 3.620875374265299*^9, 3.620875374267494*^9}, {3.620875965278901*^9, 3.620875974388295*^9}, {3.620876024298294*^9, 3.620876024298365*^9}, { 3.6208801039223633`*^9, 3.620880103924539*^9}, {3.65402800855954*^9, 3.654028008568286*^9}, {3.69705775963199*^9, 3.697057761738624*^9}}], Cell["\<\ Show first that d verifies the three properties of a distance (metric):\ \>", "SubitemParagraph", CellDingbat->None, CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208748009783773`*^9}, {3.620875123410638*^9, 3.62087512396216*^9}, 3.620875204216419*^9, {3.620875302089512*^9, 3.620875315896203*^9}, 3.620875374265299*^9, {3.620875464994075*^9, 3.620875464996376*^9}, { 3.620880936380714*^9, 3.6208809367810183`*^9}}], Cell[CellGroupData[{ Cell["d (u, v) = d (v, u)", "SubsubitemNumbered", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208748231005487`*^9}, { 3.620874866343751*^9, 3.620874866975831*^9}, {3.620875528847857*^9, 3.620875540111936*^9}}], Cell[TextData[{ "d(u, v) \[LessSlantEqual] d(u, w) + d(w, v) where u,v,w are any three \ vectors in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], "." }], "SubsubitemNumbered", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.620874887901479*^9}, { 3.620875018343892*^9, 3.620875041490272*^9}, {3.620875078593541*^9, 3.6208751974359303`*^9}, {3.620880951093967*^9, 3.620880964156722*^9}}], Cell["\<\ d(u, v) \[GreaterSlantEqual] 0 for any two vectors u,v and d(u, v) = 0 if and \ only if u = v.\ \>", "SubsubitemNumbered", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208750143439703`*^9}, { 3.620875328192192*^9, 3.620875328198288*^9}, {3.620875418656412*^9, 3.620875449323811*^9}, {3.620875636188661*^9, 3.62087563619129*^9}, { 3.620877651809614*^9, 3.620877651812574*^9}, {3.6208809530207148`*^9, 3.620880971812907*^9}, {3.65266488955429*^9, 3.652664903755641*^9}, { 3.654027988796467*^9, 3.654027988799135*^9}}] }, Open ]], Cell[TextData[{ "Next assume that u and v are two probabilistic vectors in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], ". Show that the distance function is bounded" }], "SubitemNumbered", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208748009783773`*^9}, {3.620875123410638*^9, 3.62087512396216*^9}, 3.620875204216419*^9, {3.620875302089512*^9, 3.620875315896203*^9}, { 3.620875374265299*^9, 3.620875374267494*^9}, {3.620875965278901*^9, 3.620875974388295*^9}, {3.620876024298294*^9, 3.620876024298365*^9}, { 3.6208801039223633`*^9, 3.620880103924539*^9}, 3.65402800855954*^9, 3.654028085816595*^9, {3.654028258761505*^9, 3.65402826379565*^9}}], Cell[" d(u, v) \[LessSlantEqual] 1. ", "SubsubitemNumbered", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208750143439703`*^9}, {3.620875328192192*^9, 3.620875328198288*^9}, { 3.620875418656412*^9, 3.620875449323811*^9}, {3.620875636188661*^9, 3.620875954225787*^9}, {3.6208760375296392`*^9, 3.620876100961933*^9}, { 3.620878655541204*^9, 3.6208786653229847`*^9}, {3.6208801693902273`*^9, 3.6208802736706343`*^9}, {3.620880307052762*^9, 3.620880314703223*^9}, 3.620880978943316*^9, {3.652665125288149*^9, 3.6526651386556168`*^9}, { 3.654028287550509*^9, 3.654028294250286*^9}}], Cell[TextData[{ "Assume that u and v are two probabilistic vectors in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalR]", "n"], TraditionalForm]]], ". Let S = ", StyleBox["{", FontSize->14], "1\[LessSlantEqual]i\[LessSlantEqual]n, ", Cell[BoxData[ FormBox[ SubscriptBox["u", "i"], TraditionalForm]]], "> ", Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]]], StyleBox["}", FontSize->14], " be the set of indices i for which the i-th entry of u is strictly greater \ than the j-th entry of v. " }], "SubitemParagraph", CellDingbat->None, CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208750143439703`*^9}, { 3.620875328192192*^9, 3.620875328198288*^9}, {3.620875418656412*^9, 3.620875449323811*^9}, {3.620875636188661*^9, 3.620875954225787*^9}, { 3.6208760375296392`*^9, 3.620876100961933*^9}, {3.620878655541204*^9, 3.6208786653229847`*^9}, {3.6208801693902273`*^9, 3.620880202488412*^9}, { 3.6540283088707323`*^9, 3.654028310688959*^9}, {3.6970577902683887`*^9, 3.697057791668256*^9}}], Cell[CellGroupData[{ Cell["\<\ Show that S is a proper subset of {1,2,...,n}, that is, S is nonempty and S\ \[NotEqual]{1,2,...,n}.\ \>", "SubsubitemNumbered", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208750143439703`*^9}, { 3.620875328192192*^9, 3.620875328198288*^9}, {3.620875418656412*^9, 3.620875449323811*^9}, {3.620875636188661*^9, 3.620875954225787*^9}, { 3.6208760375296392`*^9, 3.620876100961933*^9}, {3.620878655541204*^9, 3.6208786653229847`*^9}, {3.6208801693902273`*^9, 3.620880233833156*^9}, { 3.62088032210358*^9, 3.6208803237673063`*^9}, {3.620880745348159*^9, 3.620880843649631*^9}}], Cell[TextData[{ "Show that d(u, v)=", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox[ UnderscriptBox[ RowBox[{" ", "\[Sum]"}], RowBox[{"i", "\[Element]", "S"}]], FontSize->18], SubscriptBox["u", "i"]}], "-"}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]]], " and that d(u, v)<1. " }], "SubsubitemNumbered", CellDingbat->"\[FilledSquare]", CellChangeTimes->{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208750143439703`*^9}, {3.620875328192192*^9, 3.620875328198288*^9}, { 3.620875418656412*^9, 3.620875449323811*^9}, {3.620875636188661*^9, 3.620875954225787*^9}, {3.6208760375296392`*^9, 3.620876100961933*^9}, { 3.620878655541204*^9, 3.6208786653229847`*^9}, {3.6208801693902273`*^9, 3.6208802736706343`*^9}, {3.620880307052762*^9, 3.620880314703223*^9}, 3.620880978943316*^9, 3.654028327116694*^9}] }, Open ]], Cell[TextData[{ "Let A be a positive column stochastic nxn matrix. Show that there exists a \ real number r, with 0{{3.62087425935975*^9, 3.620874346730131*^9}, { 3.620874458173316*^9, 3.620874459705619*^9}, {3.6208746021388607`*^9, 3.6208746655942717`*^9}, {3.6208747331236277`*^9, 3.6208748009783773`*^9}, {3.620875123410638*^9, 3.62087512396216*^9}, 3.620875204216419*^9, {3.620875302089512*^9, 3.620875315896203*^9}, 3.620875374265299*^9, 3.620875464994075*^9, {3.6208765033573313`*^9, 3.62087653133852*^9}, {3.620876817587574*^9, 3.620877028454208*^9}, { 3.6208770907179956`*^9, 3.620877090721744*^9}, {3.620880879343355*^9, 3.62088087934573*^9}, {3.6208809855913477`*^9, 3.6208809894074*^9}, 3.620881933413995*^9, {3.654028107107326*^9, 3.654028108130517*^9}}] }, Open ]], Cell[TextData[{ StyleBox["Hint:", FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox[" ", FontColor->GrayLevel[0.5]], StyleBox["Consider the sets S={1\[LessSlantEqual]i\[LessSlantEqual]n, ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox["u", "i"], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["> ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["} and Q={1\[LessSlantEqual]i\[LessSlantEqual]n, ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{"(", "Au", ")"}], "i"], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["> ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{"(", "Av", ")"}], "i"], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["}. Use ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["Lemmas 0.1", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox[" and ", FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["0.4", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox[" to show that these are proper subsets of {1,2,...,n}. Next show \ that there exists a number r with 0"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ UnderscriptBox[ RowBox[{" ", "\[Sum]"}], RowBox[{"i", "\[Element]", "Q"}]], FontSize->18], SubscriptBox[ StyleBox["a", FontSize->16], StyleBox["ij", FontSize->10]]}], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox[" < 1-r. The number r can be chosen so that 0"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox["", RowBox[{ RowBox[{"0", "\[LessSlantEqual]", "i"}], ",", RowBox[{"j", "\[LessSlantEqual]", "n"}]}]], TraditionalForm]]], StyleBox["{", FontSlant->"Italic", FontColor->GrayLevel[0.5]], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["a", FontSize->16], StyleBox["ij", FontSize->10]], TraditionalForm]], FontSlant->"Italic", FontColor->GrayLevel[0.5]], StyleBox["}", FontSlant->"Italic", FontColor->GrayLevel[0.5]], ". 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Let v be a probabilistic \ vector of size n. 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